How do you count choices using the multiplication principle?
A small cellular provider gives its customers 2 choices of phones to use. They may use an iPhone or a phone that uses the Android operating system. In addition, the company offers three different calling plans: Budget plan, Regular plan, and the Deluxe plan. How many different choices of phone and calling plan does a customer have?
By listing out each of the possibilities, we see that there are six possible phone/plan choices. The decision tree helps us to list out these possibilities. However, if we only need to know how many choices, we can multiply the number of choices for phones and plans.
This strategy is useful for determining the total number of choices even when there are a larger number of choices.
Suppose we wish to know the number of ways to make n choices where there are
d1 ways to make choice 1
d2 ways to make choice 2
dn ways to make choice n
Then the total number of ways to make all of the choices is
d1 · d2 · … · dn
Example 1 Multiplication Principle
An online custom bicycle seller wishes to count the total number of different types of bicycles that are available through its website. The seller offers 4 different frame styles, 8 different fender colors, 10 different tire colors, 8 different wheel colors, 6 different pedal colors, and 12 different accessory colors. How many different bicycles can a customer order?
There are 184,320 different bicycles that can be ordered.
Example 2 Multiplication Principle
As the number of cars on the road has increased, so has the number of license plates. The format of the license plate determines how many different license plates there are. For each of the formats below, find the number of different license plates that are available.
a. Three numbers
b. Three letters followed by three numbers
Solution In this type of license plate, we have six choices to make. For each of the first three choices, there are 26 letters to choose from. For the last three choices, there are 10 numbers to choose from. This gives leads to the total number of license plates,
c. Six characters where each character may be a letter or number